Orthonormal time-frequency shifting and spectral shaping communications method

ABSTRACT

A wireless combination time, frequency and spectral shaping communications method that transmits data in convolution unit matrices (data frames) of N×N (N 2 ), where generally either all N 2  data symbols are received over N spreading time intervals (each composed of N time slices), or none are. To transmit, the N 2  sized data frame matrix is multiplied by a first N×N time-frequency shifting matrix, permuted, and then multiplied by a second N×N spectral shaping matrix, thereby mixing each data symbol across the entire resulting N×N matrix (TFSSS data matrix). Columns from this N 2  TFSSS data matrix are selected, modulated, and transmitted, on a one element per time slice basis. At the receiver, the replica TFSSS matrix is reconstructed and deconvoluted, revealing the data. The method can accommodate multiple users at once, can adapt to changing channel conditions, and is particularly useful for coping with channel impairments such as Doppler shifts.

This application is a continuation of U.S. patent application Ser. No.13/117,119, “ORTHONORMAL TIME-FREQUENCY SHIFTING AND SPECTRAL SHAPINGCOMMUNICATIONS METHOD”, filed May 26, 2011; U.S. patent application Ser.No. 13/117,119 in turn claimed the priority benefit of U.S. provisionalapplication 61/349,619, “ORTHONORMAL TIME-FREQUENCY SHIFTING ANDSPECTRAL SHAPING COMMUNICATIONS METHOD”, filed May 28, 2010; thecontents of both of these applications are incorporated herein byreference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention is in the general field of communications protocols andmethods, more specifically in wireless communications protocols andmethods.

2. Description of the Related Art

As wireless communications technology has advanced, the general problemof how to crowd more and more users, each wanting to send and receivemore and more amounts of data onto limited regions of the availablewireless radio spectrum has increased.

Today, there are a huge number of different portable hand-held deviceswith wireless capability in use. These portable wireless devices (suchas cell phones, portable computers, and the like) are often powered bysmall batteries, and the users typically expect these devices to operatefor many hours before the batteries are recharged. To meet these userexpectations, the wireless transmitters on these devices must outputwireless signals using very small amounts of power, making it difficultto distinguish the wireless radio signal over background noise.

An additional problem is that many of these devices are carried onmoving vehicles, such as automobiles, airplanes, and the like. Thiscauses additional complications because the low-power wireless signaltransmitted by these devices can also be subjected to variousdistortions, such as varying and unpredictable Doppler shifts, andunpredictable multi-path effects often caused by varying radioreflections off of buildings or other structures.

And against all these problems, the noise background of the variouswireless channels becomes ever higher as noise-producing electricaldevices proliferate. The proliferation of other wireless devices alsoadds to the background noise.

In order to cope with these problems, a number of prior art radiotransmission methods, including Time Division Multiple Access (TDMA),the Global System for Mobile Communications (GSM), Code DivisionMultiple Access (CDMA), Frequency Division Multiple Access (FDMA),Orthogonal Frequency-Division Multiplexing (OFDM), and other wirelessprotocols have become widespread. Each attempts to solve the above setof problems in a slightly different way.

One of the simplest, yet highly popular, wireless communications schemesis TDMA, In TDMA, a single carrier frequency can be shared with multipleusers, and each user is assigned their own particular time-slot, whichin turn is interleaved with the time slots reserved for other users.

TDMA forms the basis for the more advanced GSM method, which combinestime hopping with frequency hopping, and is the most popular mobilephone protocol in the world today. GSM allocates a number of differentfrequencies or wavelengths for use, and each wavelength is divided intosmall timeslots often only a few milliseconds in duration. Thus eachwavelength can be shared with about eight to sixteen other users.

By contrast, CDMA is more complex protocol. CDMA is a spread-spectrumprotocol that assigns each transmitter a different spreading-code. Thisspreading-code is usually a pseudo-random number or code that runs at amuch higher “chip” rate (a chip is typically a very brief rectangularpulse of +1 or −1 amplitude) than the underlying data signal that theuser wishes to transmit. CDMA works by modulating the user's data signalwith the high speed pseudo-random spreading code by, often by anexclusive OR (XOR) process, before transmitting the result. This highspeed spreading-code effectively distributes the user's data across abroader spectrum of wavelengths, and also provides a convenient way fora receiver to “tune” in to the signal and distinguish the signal frombackground noise.

By using different pseudo-random numbers, a number of different userscan use different spreading codes, and all simultaneously transmit onthe same frequency band. At the receiving end, a receiver knowing theproper pseudo-random spreading code can “tune in” to the desiredtransmitter, and ignore the other signals. The effect of the other userstransmitting using different spreading codes on the same frequency bandis minimal because their different spreading codes only raise thebackground noise on the channel by a small amount. Indeed, becausespreading codes spread the data across a broader spectral range,spread-spectrum methods such as CDMA are considered to be relativelyresistant to background noise.

By contrast, the alternative FDMA protocol works by subdividing theavailable spectrum into a number of narrow band channels. Users are eachassigned a unique frequency or wavelength (a unique narrow band) duringeach communications session.

The OFDM (also called discrete multi-tone modulation or DMT) protocolalso divides the available spectrum into many narrow band channels ortone or spectral-shapes, each separated from each other by a minimalfrequency or wavelength separation. These narrow band tone orspectral-shape channels can carry quadrature-amplitude modulated (QAM)signals or phase-shift keyed signals designed to minimally overlap (beorthogonal with) with similar signals carried by adjacent narrow bandchannels. The OFDM scheme essentially maps out a limited time-frequencygrid so that each symbol that is transmitted is allocated a particulartime and frequency coordinate. Because the OFDM scheme uses very closelypacked channels, as well as carrier modulation schemes chosen forminimal overlap between these many closely packed tone or spectral-shapechannels, the method can achieve a relatively high rate of datatransmission.

Although the OFDM scheme uses available spectrum fairly efficiently, andthus is popular for both wireless and wired wideband digitalcommunications, the method requires very accurate frequencysynchronization between the receiver and the transmitter. Thus the OFDMprotocol is relatively sensitive to problems caused by Doppler shift,and these problems are made still worse when reflections off ofbuildings and other structures cause multi-path effects.

BRIEF SUMMARY OF THE INVENTION

In view of the increasing use of wireless devices in the modern world,there is thus remains an unmet need for communications protocols thatare inherently capable of carrying wideband communications using lowamounts of power, yet are more resistant to problems of Doppler shift,multi-path reflections, and background noise than previouscommunications protocols. The invention is a new method ofcommunications intended to accomplish these goals.

Although wireless examples will be used throughout this application,note that wired communications share many of these problems as well.Thus the methods disclosed herein are intended, unless stated otherwise,to cover wired communications methods as well as wireless communicationsmethods.

The invention's method operates by spreading data across time, spectrum,waveform or tone or spectral-shape modulation using novel time-frequencyshifting and spectral shaping codes that utilize both time-shifting,frequency-shifting, and spectral shaping to a greater extent thanprevious methods. The invention's method combines both time-shiftingtechniques, frequency-shifting techniques, and spectral-shapingtechniques that, in some degenerate situations, resembles TDMA, in otherdegenerate situations resembles CDMA, and in other degenerate situationsresembles OFDM, but which in the intended use mode is its own uniquemethod, here called an Orthonormal Time-Frequency Shifting and SpectralShaping (OTFSSS, or alternatively OTFS³) method. As will be discussed,this OTFSSS method can transmit data at high rates, yet is unusuallyresistant to problems caused by Doppler shifts, multi-path effects, andbackground noise.

The invention generally operates by sending data in larger “chunks” orframes than previous methods. That is, while a prior art CDMA or OFDMmethod might encode and send N symbols over a communications link over aset interval of time, the invention will typically encode a minimum ofN² symbols over longer periods of time, and each data symbol or elementthat is transmitted is spread out to a much greater extent in time,frequency, and spectral shape space than was the case for prior artmethods. As a result the N2 symbols are assembled over a longer periodtime relative to previous methods, and it also takes longer to start toresolve the value of any given data symbol because this symbol must begradually built-up or accumulated over a longer period of time.

More specifically, during the transmitting process, the invention willsubdivide and transmit each data element or symbol over a cyclicallyvarying range of frequencies over a series of spreading time intervals.Often this will require that each data element or symbol be transmittedover a substantially longer period of time, relative to alternativecommunications protocols. In order to avoid slowing the data rate downto an unacceptably slow level, however, the invention also utilizescomplex multiplexing methods, based on various convolution anddeconvolution schemes discussed herein, to pack a comparatively largeamount of information into each signal that is sent over the cyclicallyvarying range of frequencies and over a series of spreading timeintervals. By proper selection of convolution and deconvolution schemes,the slower data rate that results from splitting a single data elementor symbol over N time-spreading intervals can be compensated for byappropriate convolution and deconvolution schemes that send the data indata frames or data matrixes [D] composed of N² data symbols orelements.

Because each data symbol is subdivided and sent over a plurality ofsignals, loss of a few of these plurality of signals during the processof transmission and reception need not result in data symbol lossbecause various signal processing schemes can be employed to compensatefor a loss of a few of the plurality of signals. At the same time,losses due to common wireless communications link impairments, such asDoppler shift and multi-path effects, can be more readily compensatedfor.

For example, whereas with prior art, if by chance Doppler effects causedby one wireless signal from a first transmitter fall on the samefrequency as another signal from a first or second transmitter (formulti-path effects, a signal from a moving first or second transmitterthat hits an object at an arbitrary angle to the receiver can produce aDoppler distorted reflection or echo signal of the first or secondtransmitter that also reaches the receiver), this could result inconfusion, ambiguity, and data loss. By contrast, according to theinvention, by cyclically shifting the frequency and sending an elementof data over a plurality of time intervals, the impact of a Doppler“collision” is substantially minimized—at most there will be a brieftransient effect resulting in the loss of only one of a plurality ofsignals used to transmit a particular data symbol or element. Theeffects of other communications link impairments, such as multi-patheffects, can also be minimized because the cyclically shifting frequencyprovides yet another way to compensate for multi-path effects.

Note that there are at least two basic ways to partition a data elementor symbol across a time range of cyclically shifting frequencies, andthus two basic forms of the invention. In the first form of theinvention, the data from a single symbol is convoluted and partitionedacross multiple time slices, and ultimately transmitted as a series oftime slices, on a per time slice basis, somewhat reminiscent of(although quite different from) TDMA schemes, and the cyclicallyshifting frequency is accomplished over a plurality of time spreadingintervals using this transmission scheme. Thus, for the first form ofthe invention, the basic unit of data transmission operates on a timeslice basis.

In the second form of the invention, the data is ultimately transmittedas a series of waveforms with characteristic frequencies, where eachwaveform lasts for a spreading interval of time generally consisting ofN time slices, and the cyclically shifting frequency is accomplishedover a plurality of time spreading intervals through this transmissionscheme. Thus, for the second form of the invention, the basic unit ofdata transmission operates over a longer spreading interval of time thatmay be viewed as generally consisting of N time slices. In general,throughout this disclosure, unless otherwise specified, the discussionwill focus on the first, time-slice oriented, form of the invention.

Returning to a discussion of the first basic form of the invention, inone embodiment, the method functions by forming a N×N data frame matrixof N² symbols or elements, multiplying this data frame by a first N×Ntime-frequency shifting matrix, optionally permuting this result, andthen multiplying the result by a second N×N spectral shaping matrix. Asa result, the N² data elements in the frame of data are essentiallymixed or distributed throughout the resulting N×N matrix product herecalled a “Time Frequency Shifted and Spectral Shaped” data matrix or“TFSSS data matrix”. Thus, for example, a single symbol or element inrow 1 column 1 of the N×N frame of data may end up being distributedover all rows and columns of the resulting N×N TFSSS data matrix.

The contents (i.e. the individual elements) of this TFSSS data matrixmay then be selected, modulated, and transmitted. Usually N elements ata time from this TFSSS data matrix (often a column from the TFSSS datamatrix) are selected to be sent over one spreading interval of time,thus often requiring N spreading intervals of time to transmit theentire contents of the TFSSS data matrix. This spreading interval oftime in turn is usually composed of at least N time slices. During eachtime slice, one element from the most recent selection of N elements(for example, from the selected column of the TFSSS data matrix) areselected, modulated, and transmitted.

At the receiving end, the process operates generally in reverse. Theindividual elements of the TFSSS data matrix are received over varioustime slices and various time spreading intervals, allowing the receiverto reassemble a replica (which may not be a perfect replica due tocommunications link impairment effects) of the original TFSSS datamatrix. Using its knowledge of the first N×N time-frequency shiftingmatrix, the optional permutation process, the second N×N spectralshaping matrix, and the selection process used to select differentelements of the TFSSS data matrix, as well as various noise reduction orcompensation techniques to overcome impairment effects, the receiverwill then reconstruct the original N×N data frame matrix of N² symbolsor elements. Because each data symbol or element from the original dataframe is often spread throughout the TFSSS data matrix, often most orall of the TFSSS matrix will need to be reconstructed in order to solvefor the original data symbol or element. However, by using noisereduction and compensation techniques, minor data losses duringtransmission can often be compensated for.

This is thus a considerable departure from prior art methods. Consider acommunications link otherwise capable of transmitting N matrix elementsper time-spreading interval. Prior art methods, such as CDMA methods orOFDM methods might transmit N matrix elements per time spreadinginterval, and after each time-spreading interval, be able to decode andcompute the underlying N data elements. By contrast, in the invention'smethod, given a communications link with the same capacity, generally Ntime spreading intervals will be required to transmit N² matrixelements, and generally most randomly selected data elements cannotdecoded before the data has been spread over N time spreading intervals,transmitted over N time spreading intervals, and then received over Ntime spreading intervals. Thus, in some OTFSSS (OTFS³) embodiments, theoverall inherent rate of data transmission (N data elements perspreading time interval) may be the same as alternative methods, but theability to access the data at intermediate times will be different.

The advantages of this approach in terms of increased ability to copewith communications link impairments will be discussed.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an overview of the OTFSSS method, here being used totransmit data over a wireless link.

FIG. 2 shows an overview of the OTFSSS method, here being used toreceive data over a wireless link.

FIG. 3 shows an example of some of the basic building blocks (basevector, data vectors, Fourier Vector and Transmit vectors) used toconvolute and deconvolute data according to the second form of theinvention.

FIG. 4 shows a diagram of the cyclic convolution method used toconvolute data and transmit data according to the second form of theinvention.

FIG. 5 shows a diagram of the cyclic deconvolution method used todeconvolute received data according to the second form of the inventionmethod.

FIG. 6 shows a transmitter following a first alternative OTFSSS scheme.

FIG. 7 shows a receiver following this first alternative OTFSSS scheme.

FIG. 8 shows a transmitter following a second alternative OTFSSS scheme.

FIG. 9 shows a receiver following this second alternative OTFSSS scheme.

FIG. 10 shows a more detailed diagram of one embodiment of an OTFSSStransmitter.

DETAILED DESCRIPTION OF THE INVENTION

A unique aspect of the invention is the concept of spreading the datafor a single symbol over a larger range of times, frequencies, andspectral shapes than has been employed by prior art methods. Althoughprevious communication schemes have used methods in which data for asingle symbol may be spread over different spectrums (i.e.spread-spectrum methods) in a given time interval, or methods in whichdata for a single element may be spread over a series of differentchannels or spectral shapes in a given time interval, or methods inwhich data for a single symbol may be assigned to be communicated to aparticular time slot, one thing that previous methods all had in commonwas that the data for any given data symbol could be assigned to aunique time-spreading interval or time slice.

By contrast, the invention is based, in part, upon the realization thatjust as there are advantages to spreading the data to communicate asingle symbol over multiple frequencies (e.g. for spread spectrummethods) or multiple spectral shapes, so too if the data to communicatea single symbol was spread over multiple time-spreading intervals aswell, there would be certain advantages.

The invention is also based, in part, upon the realization that if thedata for any given symbol is spread over time, spectrum, and spectralshapes to a greater extent than had been done previously, then thisapproach would be more resistant to interference, particularlyinterference caused by Doppler effects and multi-path effects, as wellas general background noise effects. In particular, the invention isless demanding for the need for very accurate frequency synchronizationbetween the receiver and the transmitter, which has plagued prior artmethods such as OFDM methods.

In essence, the invention convolutes the data for a group of N² symbols(herein called a “frame”) over both time, frequency, and spectral shapein a way that, because the data for each symbol is spread over a longerperiod than an otherwise equivalent prior art methods, generallyrequires that the data for the group of symbols be sent over a longerperiod of time than otherwise equivalent prior art methods. Theinvention also requires that the data for any given group of symbols beaccumulated over a longer period of time than an otherwise equivalentprior art methods.

However in one significant departure from prior art spread-spectrummethods, such as CDMA, where each different symbol might potentially betransmitted with its own unique pseudo-random spread-spectrum code, theinvention may in some embodiments compensate for the any lower data rate(that might be caused by spreading out the data for each symbol over alonger period of time) by data transmission efficiencies that resultwhen a group of symbols are transmitted using the same spread-spectrumcode. Although, using prior spread spectrum techniques, such an approachwould not be feasible because if multiple symbols are transmitted usingthe same spreading code, then confusion and ambiguity would result, theinvention may, in some embodiments, solve for this confusion andambiguity by sending the time-spread symbols using different (butpreviously defined) spread-spectrum convolution methods across thegreater number of time and frequency periods, such that when all thedata is finally accumulated at the end, the solution for the entireframe or group of symbols can then be solved for. The trade-off is thatat the end, as previously discussed generally either an entiremulti-symbol frame of data will be received, or none will be (i.e. ifthere is too much interference, then the ability to successfullydeconvolute and retrieve multiple symbols will likely fail). However aswill be discussed, the superior ability of this invention's OTFSSSmethod will generally make this trade-off acceptable.

As a simple way to contrast the differences between the invention'sOTFSSS method versus prior art CDMA, OFDM, and TDMA methods, it isimportant to realize that CDMA primarily convolutes the data signal overthe frequency domain, and OFDM primarily convolutes the data signal overa spectral shape domain, and TDMA, although not splitting the data for aparticular symbol over multiple time slots, at least has the concept ofallocating different users to different time slots. By contrast, theinvention's OTFSSS method may convolute the data signal for a singledata symbol over both a plurality of time slots, a plurality offrequencies or spectral regions (spread spectrum), and a plurality ofspectral shapes. This greater extent of data convolution results insuperior performance over impaired communications links.

Alternatively, the invention may be viewed as a method of transmittingand receiving at least one frame of data ([D]) over a communicationslink, this frame of data comprising a matrix of up to N² data elements,N being greater than 1. This method comprises convoluting the dataelements of said data frame so that the value of each data element, whentransmitted, is spread over a plurality of wireless waveforms, eachwaveform having a characteristic frequency, and each waveform carryingthe convoluted results from a plurality of said data elements from thedata frame. Further, during the transmission process, cyclicallyshifting the frequency of this plurality of wireless waveforms over aplurality of times so that the value of each data element is transmittedas a plurality of cyclically frequency shifted waveforms sent over aplurality of times. At the receiving end, receiving and deconvolutingthese wireless waveforms thereby reconstructing a replica of said atleast one frame of data [D]. Here the convolution process is such thatan arbitrary data element of an arbitrary frame of data ([D]) cannot beguaranteed to be reconstructed with full accuracy until substantiallyall of these wireless waveforms have been transmitted and received.

Matrix Formulation:

Throughout this discussion, the use of matrix terminology should beunderstood as being a concise description of the various operations thatwill be carried out by either a_(i,(j) communications link transmitteror receiver electronic circuitry (often these instructions will beimplemented by a microprocessor, digital signal processor, or other typeof electronic integrated circuit device). Thus the series of stepsrequired to obtain the coefficients of a particular matrix generallycorresponds to set of instruction for the transmitter or receiverelectronic circuitry. For example, one set of coefficients may instructthe transmitter or receiver to perform a spread spectrum operation, adifferent set of coefficients may instruct the transmitter or receiverto perform a spectral shaping modulation or demodulation operation, andanother set of coefficients may instruct the transmitter to performvarious time spreading or time accumulation functions. Here standardmatrix mathematics is used here as a shorthand way of reciting the verycomplex series of instructions needed to transmit and receive thesecomplex series of wireless signals.

Thus, when the discussion speaks of multiplying matrices, each dataelement in the matrix formed by the multiplication can be understood interms of various multi-step operations to be carried out by thetransmitter or receiver electronic circuitry, rather than as a purenumber. Thus, for example, a matrix element formed from one matrix thatmay have spread-spectrum like pseudorandom numbers times another matrixthat may have tone or spectral-shape spreading instructions, such as QAMor phase shift keying instructions, times another scanning system,permutation scheme, or matrix that may have data instructions should beunderstood as directing a transmitter to transmit a radio signal that ismodulated according to these three means, or as directing a receiver toreceive and demodulate/decode a radio signal that is modulated accordingto these three means.

Put into matrix terminology, the OTFSSS method of convoluting the datafor a group of symbols over both time, spectrum, and tone orspectral-shape can be viewed as transforming the data frame with N²information elements (symbols) to another new matrix with N² elementswhereby each element in the new transformed matrix, (here called theTFSSS data matrix) carries information about all elements of theoriginal data frame. In other words the new transformed TFSSS datamatrix carries a weighted contribution from each element of the originaldata frame matrix [D]. This TFSSS data matrix is in turn transmitted andreceived over successive time intervals.

As previously discussed, the invention's method, in which the basic unitof convolution and deconvolution (convolution unit) is composed of N²symbols or data elements, differs from prior art CDMA and OFDMapproaches, which the basic unit of convolution, transmission,reception, and deconvolution operations were focused on smallerconvolution units composed of at most only N symbols or data elements.

Additionally, over each time interval, the invention's method will use adifferent waveform for each data element. By contrast, the prior artOFDM or CDMA methods will generally always use the same waveform foreach data element.

For consistency, the N² units of data will generally be referred to inthis specification as a “data frame”. N may be any value greater thanone, and in some embodiments fall in the ranges between and including 64to 256.

Thus, for example, if the basic unit of convolution, transmission,reception and deconvolution for a prior art CDMA or OFDM communicationsprotocol is a data frame of n symbols or elements “d” operated onspreading codes that sends the data for N symbols over one spreadinginterval time where:[D _(1xn) ]=[d ₁ d ₂ . . . d _(n)]

Then by contrast, the invention will take, as the basic unit ofconvolution, transmission, reception, and deconvolution, a larger dataframe composed of N² elements or symbols “d” that, as will be discussed,will send the data for these N² elements over a plurality of spreadinginterval times (often the plurality is N).

$\left\lbrack D_{nxn} \right\rbrack = \begin{bmatrix}d_{1,1} & d_{1,2} & \ldots & d_{1,n} \\d_{2,1} & d_{2,2} & \ldots & d_{2,n} \\d_{3,1} & d_{3,2} & \ldots & d_{3,n} \\d_{4,1} & d_{4,2} & \ldots & d_{n,n}\end{bmatrix}$

In general, whenever the specification refers to a frame of data, thisshould be considered to be a reference to the N×N or N² matrix such asthe one shown above. (Note that at least some elements in the matrix canalso be zero or null elements.)

As previously discussed, the invention will spread this group of N²symbols across a communications link over multiple spreading intervalsof time (usually at least N spreading intervals or times), where eachindividual spreading interval of time is composed of at least N timeslices. Note that due to potential overhead required for synchronizationand identification purposes, in some embodiments, extra time slicesand/or extra spreading intervals of time may need to be allocated forthis overhead. Although, in most of this discussion, this overhead willgenerally be ignored, it should be understood that the disclosure isintended to also encompass methods in which such overhead exists.

The data will thus be transmitted as complex series of waveforms,usually over wireless radio signals with frequencies usually above 100MHz, and often above 1 GHz or more. These radio frequencies are thenusually received over at least n spreading time intervals, where eachspreading time interval is often composed of at least n time-slices.Once received, the original data frame will be deconvoluted (i.e. solvedfor) and the most likely coefficients of the original group of symbolsare reconstructed. It should be evident that in order to successfullydeconvolute or solve for the original data frame, the receiver willusually have prior knowledge of the time, spectrum, and tone orspectral-shape spreading algorithms used by the transmitter.

According to the invention, mathematically, the process of transmittinga data frame (or a convolution unit) of data, here expressed as an (N byN) or (N²) matrix [D], can be described using standard matrixmultiplication math as:

Transmission:

1: Construct the matrix product of a first N×N matrix [U₁] and [D](often written as either [U₁]*[D] or more simply [U1][D]—here both the“*” and simple close association (e.g. [U₁][D] both are intended torepresent matrix multiplication).

2: Optionally permute [U₁][D] by a permutation operation P. In general,any invertible permutation operation may be used. P may be an identityoperation, or alternatively may be a permutation operation thatessentially translates the columns of the original N×N [U₁][D] matrix todiagonal elements of a transformed [U₁][D]′ matrix. Thus P mayoptionally be of the form:

P creates a new N×N matrix:a′ _(i,j) =a _(i,(j−i)mod n)

Where a is the original matrix (here [U₁][D]), and a′ is the new matrix(here P([U₁][D]).

For greater clarity, in this disclosure, the result of this permutationoperation will be written as P([U₁][D]).

3: Multiply this result by a second N×N [U₂] matrix, forming:[P([U ₁ ][D])][U ₂].

And transmit this signal, according to methods to be discussed shortly.

Here [U₁] and [U₂] are both unitary N×N matrices, usually chosen tomitigate certain impairments on the (often wireless) communicationslink, such as wide band noise, narrow-band interference, impulse noise,Doppler shift, crosstalk, etc. To do this, rather than simply selecting[U₁] and [U₂] to be relatively trivial identity matrices [I], ormatrices with most of the coefficients simply being placed along thecentral diagonal of the matrix, [U₁] and [U₂] will usually be chosenwith non-zero coefficients generally throughout the matrix so as toaccomplish the desired spreading or convolution of the convolution unit[D] across spectrum and tone or spectral-shape space in a relativelyefficient and uniform manner. Usually, the matrix coefficients will alsobe chosen to maintain orthogonally or ability to distinguish between thedifferent encoding schemes embodied in the different rows of therespective matrices, as well as minimize autocorrelation effects thatcan occur when radio signals are subjected to multi-path effects.

In principle, [U₁] and [U₂] may be a wide variety of different unitarymatrices. For example, [U₁] may be a Discrete Fourier Transform (DFT)matrix and [U₂] may be a Hadamard matrix. Alternatively [U₁] may be DFTmatrix and [U₂] may be a chirp matrix. Alternatively [U₁] may be a DFTmatrix and [U₂] may also be a DFT matrix, and so on. Thus although, forpurposes of explaining certain aspects of the invention, certainspecific examples and embodiments of [U₁] and [U₂] will be given, thesespecific examples and embodiments are not intended to be limiting.

Note that a chirp matrix, [V] is commonly defined in signal processingas a matrix where, if Ψ is the chirp rate,

[V]=diag(Ψ, Ψ², . . . Ψ^(n)), Ψ=e^(jΨ), and frequency=e^(jω) where ω isthe initial center frequency of the spectrum.

Alternatively, an alternative chirp matrix may be used, filled withelements of the form:

$V_{j,k} = {\mathbb{e}}^{(\frac{{- {\mathbb{i}}}\; 2\;\pi\; k\; j^{2}}{N})}$

Where j is the matrix row, k is the matrix column, and N is the size ofthe matrix.

Other commonly used orthonormal matrices, which may be used for [U₁], or[U₂] or [U₃] (to be discussed), include Discrete Fourier matrixes,Polynomial exponent matrixes, harmonic oscillatory, matrixes, thepreviously discussed Hadamard matrixes, Walsh matrixes, Haar matrixes,Paley matrixes, Williamson matrixes, M-sequence matrixes, Legendrematrixes, Jacobi matrixes, Householder matrixes, Rotation matrixes, andPermutation matrixes. The inverses of these matrices may also be used.

As will be discussed, in some embodiments, [U₁] can be understood asbeing a time-frequency shifting matrix, and [U₂] can be understood asbeing a spectral shaping matrix. In order to preserve readability, [U₁]will thus often be referred to as a first time-frequency shiftingmatrix, and [U₂] will thus often be referred to as a second spectralshaping matrix. However use of this nomenclature is also not intended tobe limiting.

Turning to some more specific embodiments, in some embodiments, [U₁] mayhave rows that correspond to Legendre symbols, or spreading sequences,where each successive row in the matrix may be a cyclically shiftedversion of the Legendre symbols in the row above it. These Legendresymbols will occasionally also be referred to in the alternative as basevectors, and occasionally as spectrum-spreading codes.

In some embodiments, [U₂] may chosen to be a Discrete Fourier transform(DFT) matrix or an Inverse Discrete Fourier Transform matrix (IDFT).This DFT and IDFT matrix can be used to take a sequence of real orcomplex numbers, such as the N×N (P[U₁][D]) matrix, and further modulateP([U₁][D]) into a series of spectral shapes suitable for wirelesstransmission.

The individual rows for the DFT and IDFT matrix [U₂] will occasionallybe referred in the alternative as Fourier Vectors. In general, theFourier vectors may create complex sinusoidal waveforms (tone orspectral-shapes) of the type:

$X_{j}^{k} = {\mathbb{e}}^{(\frac{{{- {\mathbb{i}}}*2*\pi*j*k})}{N}}$where, for a N×N DFT matrix, X is the coefficient of the Fourier vectorin row k column N of the DFT matrix, and j is the column number. Theproducts of this Fourier vector can be considered to be tones orspectral-shapes.

Although certain specific [U₁] and [U₂] will be used to transmit anygiven data frame [D], when multiple data frames [D] are beingtransmitted, the specific [U₁] and [U₂] chosen may vary between dataframes [D], and indeed may be dynamically optimized to avoid certaincommunications link impairments over the course of transmitting manydata frames [D] over a communications session.

This process of convolution and modulation will normally be done by anelectronic device, such as a microprocessor equipped, digital signalprocessor equipped, or other electronic circuit that controls theconvolution and modulation parts of the wireless radio transmitter.Similarly the process of receiving and demodulation will also generallyrely upon a microprocessor equipped, digital signal processor equipped,or other electronic circuit that controls the demodulation,accumulation, and deconvolution parts of the wireless radio receiver.

Thus again using matrix multiplication, and again remembering that theseare all N×N matrixes, [P([U₁][D])][U₂] represents the TFSSS data matrixthat the transmitter will distribute over a plurality of time spreadingintervals, time slices, frequencies, and spectral shapes. Note againthat as a result of the various matrix operation and optionalpermutation steps, a single element or symbol from the original N×N datamatrix [D] after modulation and transmission, will be distributedthroughout the different time spreading intervals, time slices,frequencies, and spectral shapes, and then reassembled by the receiverand deconvoluted back to the original single data element of symbol.

On the receiving side, the process is essentially done in reverse. Herethe time and frequency spread replica of the TFSSS data matrix([P([U₁][D])][U₂])′ (the ‘representing the fact that this is a replica)is accumulated over multiple time spreading intervals, time slices,frequencies, and spectral shapes, and then deconvoluted by a electroniccircuit that is essentially the counterpart of the transmitting circuit,that solves for [D] by performing the operation:

Receiving:

1: Receive ([P([U₁][D])][U₂])’

2: Do a first left multiplication by the Hermetian matrix of the [U2]matrix [U₂ ^(H)], thus creating P([U₁][D]).

3: Optionally inverse permute this replica by (P([U₁][D])P⁻¹, thuscreating [U₁][D]

4. Do a second right multiplication by the Hermetian matrix of the [U₁]matrix U₁ ^(H), thus recreating [D].

Note that due to noise and other impairments in the channel, use ofinformation matrices and other standard types of noise reduction methodsto compensate for data loss or distortion due to various impairments inthe communications link will usually be done. Indeed, one largeadvantage of spreading out the original elements of the data frame [D]over so large range of times, frequencies, and spectral shapes is thatit can be readily appreciated how a loss of a few of the many differenttimes, frequencies and spectral shapes during transmission can be easilycompensated for.

Although other deconvolution methods may also be used, Hermitianmatrixes are suitable for this application because in general, for anyHermitian matrix [U] of a unitary matrix:

[U][U^(H)]=[I] where [I] is the identity matrix.

Communications links are not, of course, capable of transmitting data atan infinite rate. Here, the important data rate considerations, from theperspective of implementing the invention, is that the first N×Ntime-frequency shifting matrix, the second N×N spectral shaping matrix,and the elements of the data frame, as well as the constraints of thecommunications link (e.g. available bandwidth, power, amount of time,etc.) be chosen so that in balance (and neglecting overhead), at least Nelements of the N×N TFSSS data matrix can be transmitted over thecommunications link in one time-spreading interval. More specifically,(and again neglecting overhead) one element of the N×N TFSSS data matrixwill generally be transmitted during each time slice of eachtime-spreading interval.

Given this rate of communicating data, then typically the entire TFSSSdata matrix may be communicated over N time-spreading intervals, andthis assumption will generally be used for this discussion. However itshould be evident that given other balancing considerations between thefirst N×N time-frequency shifting matrix, the second N×N spectralshaping matrix, and the elements of the data frame, as well as theconstraints of the communications link, the entire TFSSS data matrix maybe communicated in less than N time-spreading intervals, or greater thanN time spreading intervals as well.

Generally, the contents of the TFSSS data matrix may be transmitted byselecting different elements from the TFSSS data matrix, and sendingthem over the communications link, on a one element per time slicebasis, over multiple spreading time intervals. Although in principle,this process of selecting different elements of the TFSSS data matrixcan be accomplished by a variety of different methods, such as sendingsuccessive rows of the TFSSS data matrix each single time spreadinginterval, sending successive columns of the TFSSS data matrix eachsuccessive time spreading interval, sending successive diagonals of theTFSSS data matrix each successive time spreading intervals, and so on,from the standpoint of communications link capacity, minimizinginterference, and reducing ambiguity, some schemes are often better thanothers. Thus often the [U₁] and [U₂] matrix, as well as the permutationscheme P, may be chosen to optimize transmission efficiency in responseto various impairments in the communications link.

In general, the process of transmitting (and part of the receivingprocess) the TFSSS data matrix will follow the steps of:

1: for each single time-spreading interval, selecting N differentelements of the TFSSS data matrix (often successive columns of the TFSSSmatrix will be chosen).

2: over different time slices in the given time spreading interval,selecting one element (a different element each time slice) from the Ndifferent elements of the TFSSS data matrix, modulating this element,and transmitting this element so that each different element occupiesits own time slice.3: receiving these N different replica elements of the replica TFSSSdata matrix over different said time slices in the given time spreadinginterval. Here, at the receiving end, the elements and TFSSS data willbe called replica elements and replica TFSSS matrix due to thepotentially distorting effects of various communications linkimpairments.4: demodulating these N different elements of said TFSSS data matrix.

Repeating steps 1, 2, 3, and 4 up to total of N times, therebyreassembling a replica of said TFSSS data matrix at the receiver.

In some embodiments, there may be some overhead to this basic model.Thus, for example, with some time padding (additional time slices oradditional time spreading intervals), checksums or otherverification/handshaking data, which could be transmitted in anon-convoluted manner, could be sent back by the receiver to thetransmitter on a per time-spreading interval, per N time spreadingintervals, or even on a per time slice interval in order to requestretransmission of certain parts of the TFSSS data matrix as needed

This method assumes that the receiver knows what the first N×N spreadingcode matrix [U₁] and the second N×N spectral shaping matrix [U₂] are,the permutation scheme P, as well as the particular scheme used toselect elements from the TFSSS matrix to transmit over various periodsof time. The receiver will essentially take the accumulated TFSSS datamatrix at the receiving end, and solve for the original N×N data frameusing standard linear algebra methods. Note, however, that because eachoriginal data symbol from the original data frame [D] has beenessentially “distributed” or “sprinkled” over the entire TFSSS datamatrix, and until the complete TFSSS data matrix is received by thereceiver and the solution completed, in many situations, there can be noguarantee that an arbitrary element or symbol from an arbitrary frame ofdata [D] can be reconstructed at the receiver until the number of singletime spreading intervals necessary to transmit the complete TFSSS datamatrix, which will often be N single time spreading intervals, has beenreached (or exceeded).

In order for the transmitter to successfully convolute the data frame,and the receiver then successfully deconvolute the data frame, both thefirst N×N time-frequency shifting matrix and the second N×N spectralshaping matrix should be unitary matrixes.

As previously discussed, in some embodiments, the first N×N timespreading matrix [U₁] may be constructed out of N rows of a cyclicallyshifted Legendre symbols or pseudorandom number of length N. That is,the entire N×N spreading matrix is filled with all of the various cyclicpermutations of the same Legendre symbols. In some embodiments, thisversion of the [U1] matrix can be used for spectrum spreading purposesand may, for example, instruct the transmitter to rapidly modulate theelements of any matrix that it affects rapidly over time, that is, witha chip rate that is much faster than the information signal bit rate ofthe elements of the matrix that the Legendre symbols are operating on.

In some embodiments, the second N×N spectral shaping matrix [U₂] can bea Discrete Fourier Transform (DFT) or an Inverse Discrete FourierTransform (IDFT) matrix. These DFT and IDFT matrixes can instruct thetransmitter to spectrally shift the elements of any matrix that the DFTmatrix coefficients act upon. Although many different modulation schemesmay be used, in some embodiments, this modulation may be chosen to be anOrthogonal Frequency-Division Multiplexing (OFDM) type modulation, inwhich case a modulation scheme such as quadrature amplitude modulationor phase-shift keying may be used, and this in turn may optionally bedivided over many closely-spaced orthogonal sub-carriers.

Often the actual choice of which coefficients to use for the first N×Ntime-frequency shifting matrix [U₁] and what coefficients to use forsecond N×N spectral shaping matrix [U₂] may depend on the conditions athand on the communications link. If, for example, a communications linkis suffering a particular type of impairment, such as wide band noise,narrow-band interference, impulse noise, Doppler shift, crosstalk and soon, then some first N×N time-frequency shifting matrixes and some secondN×N spectral shaping matrixes will be better able to cope with theseimpairments. In some embodiments of the invention, the transmitter andreceiver will attempt to measure these communications link impairments,and may suggest alternate types of first N×N time-frequency shiftingmatrixes [U₁] and second N×N spectral shaping matrixes to each [U₂] inorder to minimize the data loss caused by the various types ofcommunications channel impairments.

FIG. 1 shows an overview of the OTFSSS method, here being used totransmit data over a wireless link. Here a data frame (100), is an N×Nmatrix [D] containing N² symbols or data elements, is the payloadintended for transmission. In order to encode this data, the transmitterwill select an N×N [U₁] (102) matrix and an N×N [U₂] matrix (104). Aspreviously discussed, in some embodiments, the [U₁] matrix may be amatrix composed of Legendre symbols or a Hadamard matrix. This [U₁]matrix will often be designed to time and frequency shift the symbols orelements in the underlying data matrix [D].

[U₂] (104) may be a DFT or IDFT matrix often designed to spectrallyshape the signals. For example in some embodiments, [U₂] may contain thecoefficients to direct the transmitter circuits to transform the signalsover time in a OFDM manner, such as by quadrature-amplitude modulation(QAM) or phase-shift keying, or other scheme.

Usually [D] (100) will be matrix multiplied by [U₁] (102), and thematrix product of this operation [U₁][D] then optionally permuted (110)forming P([U₁][U₂]). In the next step 114), this matrix is in turnmultiplied by [U₂] (104) forming an N×N TFSS data matrix.

The various elements of the TFSSS matrix are then selected, usually acolumn of N elements at a time, on a single element at a time basis,(116) the elements are then prepared for modulation and transmission(118). During the modulation and transmission process, during each timeslice, the particular real and imaginary components of that individualTFSSS matrix element are used to control a time variant radio signal(120). This process is shown in more detail in FIG. 10).

Thus one column of the TFSSS matrix will be usually sent in one singletime spreading interval (108), where each element from this column issent in one time slice (112). Neglecting overhead effects, generally acomplete N×N TFSSS matrix can be transmitted over N single timespreading intervals (122).

FIG. 2 shows an overview of the OTFSSS method, here being used toreceive data over a wireless link. Just as the transmitter is often ahybrid analog/digital device, capable of performing matrix calculationsin the digital portion and then converting the results to analog signalsin the analog portion, so to the receiver will be capable of receivingand demodulating the radio signals in the receiver's analog portion, andthen often decoding or deconvoluting these signals in the receiver'sdigital portion. Here the original radio signals (120) are received, butare not an exact copy of the original transmitted signals because thereceived radio signals will often have additional artifacts,impairments, or distortions caused by various communications linkimpairments. Thus the impaired counterparts of original signal (120) areshown in FIG. 2 as (220). Thus replicas of the original elements of theTFSSS matrix are received and demodulated (222) every time slice (112),often one column every spreading time interval (108), and the receiverwill accumulate these elements over N single time spreading intervals(124), eventually accumulating enough elements (224) to create a replicaof the original TFSSS matrix.

In order to decode or deconvolute this TFSSS matrix, in the digitalportion of the receiver, the matrix will usually first be leftmultiplied by the Hermetian matrix of the [U₂] matrix [U₂ ^(H)] (204),and then subjected to an inverse permutation step P⁻¹ (226). Next theTFSSS matrix will be deconvoluted back to the original data matrix [D],by right multiplying (230) by the Hermetian of the original N×Ntime-frequency shifting matrix [U₁], which is [U₁ ^(H)] (202). Becauseusually the reconstructed signal will have some noise and distortion dueto various communications link impairments, usually various standardnoise reduction and statistical averaging techniques, such asinformation matrices, may be used to assist in this process (not shown).At the end of this process, the original data matrix [D] can bereconstructed.

Multiple Users

Note that this approach can also be used to simultaneously send datafrom multiple users using multiple transmitters (here usually referredto as the multiple transmitter case).

For example, assume multiple users “a”, “b”, “c”, and “d”, each sendingthe equivalent of one prior art frame of data with only N elements usingthe invention's scheme.

Each user can repackage their N elements of data into the invention'slarger N² frame of data by using a scheme in which each user packagestheir data matrix [D] according to the following scheme, where each userpacks the data into one column of their respective data frame, andleaves the other columns empty (coefficients set to zero).

User “a” sends N frames of “a” data as:

$\left\lbrack D_{a} \right\rbrack = \begin{bmatrix}a_{1,1} & 0_{1,2} & \ldots & 0_{1,n} \\a_{2,1} & 0_{2,2} & \ldots & 0_{2,n} \\\ldots & \ldots & \ldots & \ldots \\a_{n,1} & 0_{n,2} & \ldots & 0_{n,n}\end{bmatrix}$User “b” sends N frames of “b” data as

$\left\lbrack D_{b} \right\rbrack = \begin{bmatrix}0_{1,1} & b_{1,2} & \ldots & 0_{1,n} \\0_{2,1} & b_{2,2} & \ldots & 0_{2,n} \\\ldots & \ldots & \ldots & \ldots \\0_{n,1} & b_{n,2} & \ldots & 0_{n,n}\end{bmatrix}$And user “n” sends N frames of “n” data as

$\left\lbrack D_{n} \right\rbrack = \begin{bmatrix}0_{1,1} & 0_{1,2} & \ldots & n_{1,n} \\0_{2,1} & 0_{2,2} & \ldots & n_{2,n} \\\ldots & \ldots & \ldots & \ldots \\0_{n,1} & 0_{n,2} & \ldots & m_{n,n}\end{bmatrix}$

Since each independent user “a”, “b” . . . “n” can know what theirdesignated slot on the data matrix is, then all can transmit on the samechannel (same communications link). The signals will sum up gracefully,and the various signals [D_(a)], [D_(b)] . . . [D_(a)] can be receivedat the receiver as if it was a complete data frame sent by only onetransmitter, and deconvoluted as before.

As previously discussed, one advantage of the invention's approach isincreased resistance to Doppler shifts and frequency shifts. Forexample, consider what would happen if the signals transmitted by any ofthe different users “a”, “b”— . . . “n” are frequency shifted. If aprior art wireless transmission method, such as OFDM was used, thefrequency shifts and time shifts would cause a substantial amount ofinterference with other wireless devices that are transmitting andreceiving in neighboring areas. By contrast, by using the invention'sOTFSSS method, only the wireless devices transmitting on adjacent (i.e.on a local row or column on the matrix) regions of the TFSSS data matrixto the affected wireless device will be affected.

In many cases, due to the invention's greater degree of time, frequency,and spectral shaping, the net effect of the impaired device will belargely transparent due to the invention's superior ability to functionover an impaired communications link. In other cases, because the localimpaired device can be identified with greater accuracy, the basestation can either send corrective signals to the impaired device, oralternatively shut off the impaired device.

Returning to the specific case where [U₁] may have rows that correspondto pseudo-random sequences, it may be useful to employ a scheme whereeach successive row in the matrix is a cyclically rotated version of thepseudo-random sequence in the row above it. Thus the entire N×N matrixmay consist of successive cyclically rotated versions of a singlepseudo-random sequence of length N.

In an alternative embodiment, after the matrix multiplication steps, thesystem may then transmit the diagonals of the resulting matrix over aseries of single time-spreading intervals, one diagonal per singletime-spreading interval, so that N (or more generally N data elements)columns of the final N by N matrix are transmitted over N timeintervals.

The order in which the individual elements of the TFSSS data matrix[[U₁][D]][U₂] are transmitted across the communications link canalternatively be controlled by a Transmit Matrix or Transmit Vector.

For FIGS. 3, 4, and 5, we will briefly depart from the discussion of thefirst form of the invention, in which data is transmitted on a per timeslice basis, and briefly discuss the second form of the invention, inwhich data is transmitted by an alternative process in which theunderlying unit of transmission is not in the form of brief pulses on atime slice basis, but rather as a series of waveforms that last for aspreading interval of time, that is generally over N time slices.

The general concept for the second form of the invention is thatalthough it also is a method of transmitting and receiving at least oneframe of data [D] (composed of a matrix of up to N² data elements) overa communications link, in this second form, each data element isassigned a unique waveform (designated a corresponding waveform) whichis derived from a basic waveform of duration N time slices. However likethe first form of the invention, the second form of the invention alsohas the net effect of spreading the data elements of the data matrix [D]over a range of times and cyclic frequency shifts using a differentunderlying process.

In the second form of the invention, each data element is assigned aunique waveform (corresponding waveform) which is derived from a basicwaveform of length N time slices, with a data element specificcombination of a time and frequency cyclic shift of this basic waveform.

In the second form of the invention, each element in the frame of data[D] is multiplied by its corresponding waveform, producing a series ofN² weighted unique waveforms. Over one spreading time interval(generally composed of N time slices), all N² weighted unique waveformscorresponding to each data element in said fame of data [D] aresimultaneously combined and transmitted. Further, a different uniquebasic waveform of length (or duration) of N time slices is used for eachconsecutive time-spreading interval. wherein a different unique basicwaveform of length N time slices is used for each consecutivetime-spreading interval, and this set of N unique waveforms form anorthonormal basis. Essentially, each element of [D] is transmitted (inpart) again and again over all N time spreading intervals.

To receive data using this second form of the invention, over eachspreading interval of time (again composed of N time slices), thereceived signal is correlated with the set of all N² waveformspreviously assigned to each data element by said transmitter for thatspecific time spreading interval. (Thus just like otherencoding/decoding methods, this second form of the invention alsorequires that the receiver have knowledge of the set of N² waveformsthat the transmitter will assign to each data element.). Upon performingthis correlation, the receiver will produce a unique correlation scorefor each one of the N² data elements. This process will be repeated overall N time-spreading intervals. The original data matrix [D] can thus bereconstructed by the receiver by, for each data element, summing thecorrelation scores over N time-spreading intervals, and this summationof the correlation scores will reproduce the N² data elements of theframe of data [D].

FIG. 3 shows an example of some of the basic building blocks (basevector, data vectors, Fourier Vector and Transmit vectors) used toconvolute and deconvolute data according to the second form of theinvention. Here the data vector (300) can be understood as being Nelements (often one row, column, or diagonal) of the N×N [D] matrix, thebase vector (302) can be understood as being N elements (often one row,column, or diagonal) of the N×N [U₁] matrix, the Fourier vector (304)can be understood as being N elements (often one row, column, ordiagonal) of the N×N [U₂] matrix often a DFT or IDFT matrix. Thetransmit vector (306) can be understood as controlling the scanningprocess, and the transmit frame (308) is composed of units Tm (310) eachof which is essentially a spreading time interval, itself composed of aplurality of time slices. Thus the transmit vector can be understood ascontaining N single time-spreading intervals (122) (310), which in turnare composed of multiple (such as N) time slices.

Note the difference in wireless radio signal modulation between thissecond form of the invention, where each waveform exists over a timespreading interval composed of multiple (e.g. N) time slices, and thefirst form of the invention, where the wireless signal is essentiallytransmitted on a per time slice basis. This is represented by lines(312) showing that each Fourier vector waveform (304) is manifested overthe spreading time interval T^(m) (310).

FIG. 4 shows a diagram of a cyclic convolution method that may be usedto convolute data and transmit data according to the second form of theinvention. As previously discussed, particularly in the case where [U₁]is composed of a cyclically permuted Legendre number of length N, thenthe matrix math process of convoluting the data and scanning the datacan be understood alternatively as being a cyclic convolution of theunderlying data. Here the d⁰, d^(k),d^(N-1) can be understood as beingthe elements or symbols of the data vector (300) component of the [D]matrix, the b^(m) coefficients can be understood as representing thebase vector (302) components of the [U₁] matrix, and the X coefficientscan be understood as representing the Fourier vector (304) components ofthe [U₂] matrix.

FIG. 5 shows a diagram of a cyclic deconvolution method that may be usedto deconvolute received data according to the second form of theinvention. Again, as previously discussed, particularly in the casewhere [U₁] is composed of a cyclically permuted Legendre number oflength N, then the matrix math process of deconvoluting the data andreconstructing the data can be understood alternatively as being acyclic deconvolution of the transmitted data previously convoluted inFIG. 4. Here the ˜d⁰, ˜d^(k), ˜d^(N-1) can be understood as being thereconstructed elements (symbols) of the data vector (400) component ofthe [D] matrix, the b^(m) coefficients again can be understood asrepresenting the base vector (302) components of the [U₁] matrix, andthe X coefficients can again be understood as representing the Fouriervector (304) components of the [U₂] matrix. Here (R_(m)) (402) is aportion of the accumulated signal (230) received and demodulated by thereceiver.

In this alternative scheme or embodiment, the invention can beunderstood as being a method of transmitting at least one frame of data([D]) over a communications link, comprising: creating a plurality oftime-spectrum-tone or spectral-shape spreading codes operating over aplurality of time-spreading intervals, each single time-spreadinginterval being composed of at least one clock intervals; eachtime-spectrum-tone or spectral-shape spreading code comprising afunction of a first time-frequency shifting, a second spectral shaping,and a time spreading code or scanning and transmission scheme.

If the single time-spreading interval was only composed of onetime-slice or clock interval, then the system could be viewed asessentially degenerating into a TDMA-like transmission method. If thetime-frequency shifting was a constant, then the system could be viewedas essentially degenerating into an OFDM-like method. If the spectralshaping was a constant and the time-spreading interval was a pluralityof clock intervals, then the system could be viewed as essentiallydegenerating into a CDMA-like method.

Returning to a discussion of the first form of the invention, somevariations on the basic [[U₁][D]][U₂] data transmission andreconstruction process are also possible, and these variations are shownin more detail in FIGS. 6 to 9.

FIG. 6 shows a first transmitter scheme following a first alternativescheme. Here the data matrix [D] may be further convoluted by means of athird unitary matrix [U₃], which may be an IDFT matrix. Here [U1] may bea DFT matrix, and [U₂] may be the product of a DFT matrix times a base.In this scheme, the process of scanning and transmitting the data isrepresented by the previously described permutation operation P. Thebasic transmission process can thus be represented as[U₃]*[P([U₁][D])]*[U₂]. Here [D] is shown as (600), and ([U₁][D]) isshown as (602). The permuted version of this, P([U₁][D]) is shown as(604). [U₃] is shown as (606), [U₂] is shown as (608), and the finalproduct [U₃][P([U₁][D])][U₂] is shown as (610). Note that [U3] can be aDFT matrix, an IDFT matrix, or a trivial identity matrix (in which casethis first alternative scheme reduces back to schemes where [U3] is notinvoked at all).

Some of the effects of the permutation operation P are symbolized by thechanging location of the arrow from the column of (602) to the diagonalof (604).

FIG. 7 shows a receiver following this first alternative scheme. Herethe data that the transmitter has received and accumulated, aftervarious communications link impairment effects, is represented as the[r] matrix. The [r] matrix is demodulated and deconvoluted (decoded) byforming the Hermetian matrices of the original [U₁], [U₂], and [U₃]matrices originally used to encode and modulate the data [D], as well asthe inverse permutation operation P⁻¹ to undo the original permutationoperation P used to scan and transmit the data over multiple timeintervals. Here [U₁ ^(H)] may be an IDFT matrix, [U₃ ^(H)] may be a DFTmatrix, and [U₂ ^(H)] may be a DFT matrix times a base.

Here [r] is (700), [U₂ ^(H)] is (702), and P⁻¹ ([U₃ ^(H)][r][U₂ ^(H)])is (704), and the reconstructed data matrix [D] is (706).

FIG. 8 shows a transmitter following a second alternative scheme. Herethe original data matrix [D] is (800), [U₁][D] is (802), and P([U₁][D])is (804). [U₂] is (806), and the transmitted signal, which is[P([U₁][D])]*[U₂] is (808). Here [U₁] may be a Hadamard matrix (i.e., asquare matrix composed of mutually orthogonal rows and either +1 or −1coefficients. This matrix has the property that H*H^(T)=nI_(n) whereI_(n) is an n×n identity matrix and H^(T) is the transpose of H).

Some of the effects of the permutation operation P are symbolized by thechanging location of the arrow from the column of (802) to the diagonalof (804).

FIG. 9 shows a receiver following this second alternative scheme. Here[r] again symbolizes the received data. As before, the [r] matrix isdemodulated and deconvoluted (decoded) by forming the Hermetian matricesof the original [U₁], and [U₂], matrices originally used to encode andmodulate the data [D], as well as the inverse permutation operation P⁻¹to undo the original permutation operation P used to scan and transmitthe data over multiple time intervals. Here [r] is (900), [U₂ ^(H)] is(902), P⁻¹([r][U₂ ^(H)]) is (904) and the reconstructed data matrix [D](created from [U₁ ¹¹]*P⁻¹([r]*[U₂ ^(H)])) is (906).

FIG. 10 shows a more detailed example of one embodiment of an OTFSSStransmitter (1000), previously shown in less detail in FIG. 1. Thistransmitter can roughly be broken down into a more digitally orientedcomputation end (1002) and a more analog signal oriented modulation end(1004). At the digital end (1002), a electronic circuit, which may be amicroprocessor, digital signal processor, or other similar device willaccept as input the data matrix [D] (100) and may either generate oraccept as inputs the [U₁] (102) and [U₂] (104) matrices as well as theoptional permutation scheme P. The digital section will then generatethe TFSSS matrix. Once generated, individual elements from this matrixwill be selected, often by first selecting one column of N elements fromthe TFSSS matrix, and then scanning down this column and picking outindividual elements at a time (1006). Generally one new element will beselected every time slice (112).

Thus every successive time slice, one element from the TFSSS matrix(1008) will be used to control the modulation circuit (1004). In oneembodiment of the invention, the modulation scheme will be one where theelement will be separated into its real and imaginary components,chopped and filtered, and then used to control the operation of a sinand cosine generator, producing an analog radio waveform (120). Thiswaveform then travels to the receiver where it is demodulated anddeconvoluted as previously shown in FIG. 2. Thus in this scheme (againneglecting overhead effects), element t_(1,1) from the first column ofthe TFSSS matrix can be sent in the first time slice, and the nthelement from the first column of the TFSSS matrix can be sent in thelast time slice of the first time spreading interval. The next t_(1,2)element from the second column of the TFSSS matrix can be sent in thefirst time slice of the second time spreading interval, and so on.

Accommodating Multiple Users

As previously discussed, multiple users that are using differenttransmitters (or simply multiple transmitters) may communicate over thesame communications link using the same protocol. Here, each user ortransmitter may, for example, select only a small number of dataelements in the N² sized frame of data to send or receive theirrespective data. As one example, a user may simply select one column ofthe frame of data for their purposes, and set the other columns at zero.The user's device will then compute TFSSS data matrices and send andreceive them as usual.

Accommodating Different Types of Communications Link (Channel)Impairments

As previously discussed, one advantage of the invention's method isincreased resistance to communications channel impairments. Thisresistance to impairments can be improved by further selecting the firstN×N time-frequency shifting matrix and the second N×N spectral shapingmatrix are selected to minimize the impact of a aberrant transmittersuffering from Doppler shift or frequency shift on the elements of theTFSSS data matrix that are adjacent to the elements of said TFSSS datamatrix occupied by said aberrant transmitter. Alternatively, thereceiver may analyze the problem, determine if an alternate set of firstN×N time-frequency shifting matrices and/or said second N×N spectralshaping matrices would reduce the problem, and suggest or command thatcorresponding changes be made to corresponding the transmitter(s).

Power and energy per symbol considerations:

Relative to prior art, the invention also enables more sophisticatedtradeoffs between transmission distance, transmitter power, andinformation data rate. This is because each symbol is spread over moreintervals. For example, for TDMA, the power per symbol transmitted mustbe quite high because the symbol is being transmitted over only one timeinterval. For CDMA, the symbol is being transmitted over essentially Nintervals, and the power per interval is correspondingly less. Becausethe invention transmits a bit or symbol of information over N² differentmodalities (e.g. waveforms, times), the power per modality is much less.Among other things, this means that the effect of impulse noise, thatwould in general only impact a specific waveform over a specific timeinterval, will be less. It also means that due to increased number ofsignal transmission modalities (waveforms, times) enabled by theinvention, there are more degrees of freedom available to optimize thesignal to best correspond to the particular communications linkimpairment situation at hand.

The invention claimed is:
 1. A method of transmitting and receiving atleast one frame of data ([D]) over a communications link, said frame ofdata comprising a matrix of up to N² data elements, N being greater than1, said method comprising: obtaining an orthonormal matrix set, saidorthonormal matrix set comprising a first N×N matrix ([U₁]) and a secondN×N matrix ([U₂]); wherein said communications link and said orthonormalmatrix set are chosen to be capable of transmitting at least N elementsfrom a matrix product of said first N×N matrix ([U₁]), a frame of data([D]), and said second N×N matrix ([U₂]) over one time spreadinginterval; said time spreading interval consisting of at least N timeslices; forming a first matrix product of said first N×N matrix (pa andsaid frame of data ([D]); permuting said first matrix product by aninvertible permutation operation P forming a permuted first matrixproduct P([U₁][D]); forming a second matrix product of said permutedfirst matrix product P([U₁][D]) and said second matrix ([U₂]) forming aconvoluted data matrix; transmitting, using a hybrid analog and digitaltransmitter, and receiving, using a hybrid analog and digital wirelessreceiver, said convoluted data matrix over said communications link bythe steps of: 1: for each single time-spreading interval, selecting Ndifferent elements of said convoluted data matrix; 2: over differentsaid time slices in said time spreading interval, using said transmitterto perform the process of selecting said N different elements of saidconvoluted data matrix, modulating said N different elements, andtransmitting said N different elements so that each said N differentelements are transmitted over said time spreading interval; 3: usingsaid receiver to perform the process of receiving said N differentelements of said convoluted data matrix over different said time slicesin said time spreading interval; 4: demodulating said N differentelements of said convoluted data matrix; repeating steps 1, 2, 3, and 4up to total of N times, thereby reassembling a replica of saidconvoluted data matrix at said receiver; using said receiver, said firstN×N matrix ([U₁]) and said second N×N matrix ([U₂]) to reconstruct saidframe of data ([D]) from said convoluted data matrix; and wherein anarbitrary data element of an arbitrary frame of data ([D]) cannot beguaranteed to be reconstructed with full accuracy until a substantiallycomplete replica of said convoluted data matrix has been recovered. 2.The method of claim 1, wherein said first matrix ([U₁]) and said secondmatrix ([U₂]) are selected from the group consisting of 1: said firstmatrix ([U₁]) is a DFT matrix and said second matrix ([U₂]) is aHadamard matrix, 2: said first matrix ([U₁]) is a DFT matrix and saidsecond matrix ([U₂]) is a chirp matrix, 3: said first matrix ([U₁]) is aDFT matrix and said second matrix ([U₂]) is a DFT matrix.
 3. The methodof claim 1, wherein said first N×N matrix ([U_(i)]) is a time-frequencyshifting matrix ([U₁]), and said second N×N matrix ([U₂]) is aspectral-shaping matrix ([U₂]).
 4. The method of claim 1, wherein afterforming a first matrix product of said first N×N matrix ([U₁]), and saidframe of data ([D]), and permuting said first matrix product by aninvertible permutation operation P thereby, forming a permuted firstmatrix product P([U₁][D]) further following the steps comprising: atsaid transmitter, using a third unitary N×N matrix ([U₃]), which is aDFT matrix or an IDFT matrix, to matrix multiply said permuted firstmatrix product P([U₁][D]); and at said receiver using said third unitarymatrix ([U₃]) to reconstruct said frame of data ([D]).
 5. A method oftransmitting and receiving at least one frame of data ([D]) over acommunications link, said frame of data comprising a matrix of up to N²data elements, N being greater than 1, said method comprising: obtainingan orthonormal time-frequency shifting and spectral shaping matrix set,said orthonormal time-frequency shifting and spectral shaping matrix setcomprising a first N×N time-frequency shifting matrix ([U₁]) and asecond N×N spectral-shaping matrix ([U₂]); wherein said communicationslink and said orthonormal time-frequency shifting and spectral-shapingmatrix set are chosen to be capable of transmitting at least N elementsfrom a matrix product of said first N×N time-frequency shifting matrix([U₁]), a frame of data ([D]), and said second N×N spectral shapingmatrix ([U₂]) over one time spreading interval; said time spreadinginterval consisting of at least N time slices; forming a first matrixproduct of said first N×N time-frequency shifting matrix ([U₁]), andsaid frame of data ([D]); permuting said first matrix product by aninvertible permutation operation P forming a permuted first matrixproduct P([U₁][D]); forming a second matrix product of said permutedfirst matrix product P([U₁][D]) and said spectral-shaping matrix ([U₂])forming a time-frequency-shifted spectral-shaping data matrix (TFSSSdata matrix); transmitting, using a hybrid analog and digitaltransmitter, and receiving, using a hybrid analog and digital wirelessreceiver, said TFSSS data matrix over said communications link by thesteps of: 1: for each single time-spreading interval, selecting Ndifferent elements of said TFSSS data matrix; 2: over different saidtime slices in said time spreading interval, using said transmitter toperform the process of selecting said N different elements of said TFSSSdata matrix, modulating said N different elements, and transmitting saidN different elements so that each said N different elements aretransmitted over said time spreading interval; 3: using said receiver toperform the process of receiving said N different elements of said TFSSSdata matrix over different said time slices in said time spreadinginterval; 4: demodulating said N different elements of said TFSSS datamatrix; repeating steps 1, 2, 3, and 4 up to total of N times, therebyreassembling a replica of said TFSSS data matrix at said receiver; usingsaid receiver, said first N×N time-frequency shifting matrix ([U₁]) andsaid second N×N spectral shaping matrix ([U₂]) to reconstruct said frameof data ([D]) from said TFSSS data matrix; and wherein an arbitrary dataelement of an arbitrary frame of data ([D]) cannot be guaranteed to bereconstructed with full accuracy until a substantially complete replicaof said TFSSS data matrix has been recovered.
 6. The method of claim 5,in which said first N×N time-frequency shifting matrix ([U₁]) is aDiscrete Fourier Transform (DFT) matrix, and the second N×Nspectral-shaping matrix ([U₂]) is composed of cyclically shiftedLegendre symbols.
 7. The method of claim 5, in which said first N×Ntime-frequency shifting matrix ([U₁]) is composed of cyclically shiftedLegendre symbols, and the second N×N spectral-shaping matrix ([U₂]) is aDiscrete Fourier Transform (DFT) matrix.
 8. The method of claim 5,wherein said time-frequency shifting matrix ([U₁]) and saidspectral-shaping matrix ([U₂]) are selected from the group consisting of1: said time frequency shifting matrix ([U₁]) is a DFT matrix and saidspectral-shaping matrix ([U₂]) is a Hadamard matrix, 2: saidtime-frequency shifting matrix ([U₁]) is a DFT matrix and saidspectral-shaping matrix ([U₂]) is a chirp matrix, 3: saidtime-frequency-shifting matrix ([U₁]) is a DFT matrix and saidspectral-shaping matrix ([U₂]) is a DFT matrix.
 9. The method of claim5, wherein said orthonormal time-frequency shifting and spectral shapingmatrix set are chosen based upon communications link conditions.
 10. Themethod of claim 5, in which a plurality of different transmitters maycommunicate over said communications link by each selecting different ornon-overlapping data elements of said frame of data to use to transmitor receive their respective data.
 11. The method of claim 10, whereinsaid first N×N time-frequency shifting matrix ([U₁]) and said second N×Nspectral-shaping matrix ([U₂]) are selected to minimize the impact of anaberrant transmitter suffering from Doppler shift or frequency shift onadjacent elements to the elements of said TFSSS data matrix occupied bysaid aberrant transmitter.
 12. The method of claim 10, in which atransmitter suffering from Doppler shift or frequency shift is detectedby at least one receiver and instructed to correct for said Dopplershift or said frequency shift.
 13. The method of claim 5, in which saidfirst N×N time-frequency shifting matrix ([U₁]), is a matrix ofOrthogonal Frequency Division Multiplexing (OFDM) spreading codes, andsaid second N×N spectral-shaping-matrix ([U₂]) is a matrix of CodeDivision Multiple Access (CDMA) spreading codes.
 14. The method of claim5, in which said permutation operation P is an identity operation. 15.The method of claim 5, in which selecting N different elements of saidTFSSS data matrix is done by selecting said N different elements fromthe columns of said TFSSS data matrix.
 16. A method of transmitting atleast one frame of data ([D]) over a communications link, said frame ofdata comprising a matrix of up to N² data elements, N being greater than1, said method comprising: obtaining an orthonormal matrix set, saidorthonormal matrix set comprising a first N×N matrix ([U₁]) and a secondN×N matrix ([U₂]); wherein said communications link and said orthonormalmatrix set are chosen to be capable of transmitting at least N elementsfrom a matrix product of said first N×N matrix ([U₁]) a frame of data([D]), and said second N×N matrix ([U₂]) over one time spreadinginterval; said time spreading interval consisting of at least N timeslices; forming a first matrix product of said first N×N matrix (pa andsaid frame of data ([D]); permuting said first matrix product by aninvertible permutation operation P forming a permuted first matrixproduct P([U₁][D]); forming a second matrix product of said permutedfirst matrix product P([U₁][D]) and said second matrix ([U₂]), thusforming a convoluted data matrix; transmitting, using a hybrid analogand digital transmitter, and receiving, using a hybrid analog anddigital wireless receiver, said convoluted data matrix over saidcommunications link by the steps of: 1: for each single time-spreadinginterval, selecting N different elements of said convoluted data matrix;2: over different said time slices in said time spreading interval,selecting said N different elements of said convoluted data matrix,modulating said N different elements, and transmitting said N differentelements so that each said N different elements are transmitted oversaid time spreading interval; repeating steps 1, and 2, up to total of Ntimes, until a substantially complete replica of said convoluted datamatrix has been transmitted.
 17. The method of claim 16, wherein saidhybrid analog and digital wireless transmitter comprises amicroprocessor or a digital signal processor.
 18. A method of receivingat least one frame of data ([D]) over a communications link, said frameof data comprising a matrix of up to N² data elements, N being greaterthan 1, said method comprising: obtaining an orthonormal matrix set anda permutation operation P used by at least one transmitter to transmitsaid at least one frame of data [D], said orthonormal matrix setcomprising a first N×N matrix ([U₁]) and a second N×N matrix ([U₂]);said orthonormal matrix set is chosen to be capable of receiving atleast N elements from a matrix product of said first N×N matrix ([U₁]),a frame of data ([D]), and said second N×N matrix ([U₂]) over one timespreading interval; said time spreading interval consisting of at leastN time slices; receiving, using a hybrid analog and digital wirelessreceiver, using said receiver, a replica of a convoluted[P([U₁][D])][U₂] data matrix over said communications link by the stepsof: 1: for each single time-spreading interval, receiving transmittedsignals corresponding to substantially all of said N elements; 2:demodulating substantially all of said N different elements; repeatingsteps 1, and 2 up to total of N times, thereby reassembling a replica ofsaid convoluted [P([U₁][D])][U₂] data matrix at said receiver; usingsaid permutation operation P, said first N×N matrix ([U₁]) and saidsecond N×N matrix ([U₂]) to reconstruct said frame of data ([D]) fromsaid replica of a convoluted [P([U₁][D])][U₂] data matrix; and whereinan arbitrary data element of an arbitrary frame of data ([D]) cannot beguaranteed to be reconstructed with full accuracy until a substantiallycomplete replica of a convoluted [P([U₁][D])][U₂] data matrix has beenrecovered.
 19. The method of claim 18, wherein said hybrid analog anddigital wireless receiver comprises a microprocessor or a digital signalprocessor.
 20. The method of claim 1, wherein said N different elementsare transmitted over said time spreading interval as complex sinusoidalshaped waveforms described by a Fourier vector; wherein said first N×Nmatrix ([U₁]) comprise rows corresponding to at least one of Legendresymbols or spreading sequences, and wherein each successive row is acyclically shifted version of said at least one of Legendre symbols orspreading sequences in the row above it; wherein for said transmission,said second N×N matrix ([U₂]) comprise either a Discrete FourierTransform Matrix (DFT) or Inverse Discrete Fourier Transform (IDFT);further using said N×N matrix ([U₂]) and said hybrid analog and digitalwireless transmitter to modulate said permuted first matrix productP([U₁][D]) into a series of complex sinusoidal waveforms of the type${X_{j}^{k} = {\mathbb{e}}^{(\frac{{{- {\mathbb{i}}}*2*\pi*j*k})}{N}}};$where X is the coefficient of said Fourier vector in row k of saidsecond N×N matrix ([U2]), and j is the column number.
 21. The method ofclaim 5, wherein said N different elements are transmitted over saidtime spreading interval as complex sinusoidal shaped waveforms describedby a Fourier vector; wherein said first N×N matrix ([U₁]) comprise rowscorresponding to at least one of Legendre symbols or spreadingsequences, and wherein each successive row is a cyclically shiftedversion of said at least one of Legendre symbols or spreading sequencesin the row above it; wherein for said transmission, said second N×Nmatrix ([U₂]) comprise either a Discrete Fourier Transform Matrix (DFT)or Inverse Discrete Fourier Transform (IDFT); further using said N×Nmatrix ([U₂]) and said hybrid analog and digital wireless transmitter tomodulate said permuted first matrix product P([U₁][D]) into a series ofcomplex sinusoidal waveforms of the type${X_{j}^{k} = {\mathbb{e}}^{(\frac{{{- {\mathbb{i}}}*2*\pi*j*k})}{N}}};$where X is the coefficient of said Fourier vector in row k said ofsecond N×N matrix ([U2]), and j is the column number.
 22. The method ofclaim 16, wherein said N different elements are transmitted over saidtime spreading interval as complex sinusoidal shaped waveforms describedby a Fourier vector; wherein said first N×N matrix ([U₁]) comprise rowscorresponding to at least one of Legendre symbols or spreadingsequences, and wherein each successive row is a cyclically shiftedversion of said at least one of Legendre symbols or spreading sequencesin the row above it; wherein for said transmission, said second N×Nmatrix ([U₂]) comprise either a Discrete Fourier Transform Matrix (DFT)or Inverse Discrete Fourier Transform (IDFT); further using said N×Nmatrix ([U₂]) and said hybrid analog and digital wireless transmitter tomodulate said permuted first matrix product P([U₁][D]) into a series ofcomplex sinusoidal waveforms of the type${X_{j}^{k} = {\mathbb{e}}^{(\frac{{{- {\mathbb{i}}}*2*\pi*j*k})}{N}}};$where X is the coefficient of said Fourier vector in row k of saidsecond N×N matrix ([U2]), and j is the column number.
 23. The method ofclaim 18, wherein said N different elements were previously transmittedby a hybrid analog and digital wireless transmitter over said timespreading interval as complex sinusoidal shaped waveforms described by aFourier vector; wherein said first N×N matrix ([U₁]) comprised rowscorresponding to at least one of Legendre symbols or spreadingsequences, and wherein each successive row was a cyclically shiftedversion of said at least one of Legendre symbols or spreading sequencesin the row above it; wherein for said previous transmission, said secondN×N matrix ([U₂]) comprised either a Discrete Fourier Transform Matrix(DFT) or Inverse Discrete Fourier Transform (IDFT); wherein saidprevious transmission further used said N×N matrix ([U₂]) and saidhybrid analog and digital wireless transmitter to modulate said permutedfirst matrix product P([U₁][D]) into a series of complex sinusoidalwaveforms of the type${X_{j}^{k} = {\mathbb{e}}^{(\frac{{{- {\mathbb{i}}}*2*\pi*j*k})}{N}}};$where X was the coefficient of said Fourier vector in row k of saidsecond N×N matrix ([U2]), and j was the column number.
 24. The method ofclaim 1, wherein between each repeating step of transmitting said Ndifferent elements over said time spreading interval, further allocatingextra time slices before repeating.
 25. The method of claim 5, whereinbetween each repeating step of transmitting said N different elementsover said time spreading interval, further allocating extra time slicesbefore repeating.
 26. The method of claim 16, wherein between eachrepeating step of transmitting said N different elements over said timespreading interval, further allocating extra time slices beforerepeating.
 27. The method of claim 18, wherein between each repeatingstep of transmitting said N different elements over said time spreadinginterval, further allocating extra time slices before repeating.